On Complete Gradient Steady Ricci Solitons with Vanishing D-tensor

TL;DR

This paper classifies complete gradient steady Ricci solitons with vanishing D-tensor, showing they are either Ricci-flat or isometric to the Bryant soliton, extending previous work to broader cases.

Contribution

It extends classification results of gradient Ricci solitons with vanishing D-tensor to include steady, shrinking, and expanding cases for dimensions n ≥ 5.

Findings

Complete noncompact gradient steady Ricci solitons with vanishing D-tensor are Ricci-flat or Bryant solitons.

The classification extends to shrinking and expanding Ricci solitons.

The results generalize previous classifications to higher dimensions and broader cases.

Abstract

In this paper, we extend the work of Cao-Chen [9] on Bach-flat gradient Ricci solitons to classify $n$-dimensional ($n\ge 5$) complete $D$-flat gradient steady Ricci solitons. More precisely, we prove that any $n$-dimensional complete noncompact gradient steady Ricci soliton with vanishing $D$-tensor is either Ricci-flat, or isometric to the Bryant soliton. Furthermore, the proof extends to the shrinking case and the expanding case as well.